An elementary approach to component sizes in critical random graphs

TL;DR

This paper introduces a straightforward method to bound the probability of large components in critical random graphs, applicable to various models including intersection graphs, percolation on regular graphs, and inhomogeneous graphs.

Contribution

The paper presents a simple, unified approach to derive polynomial bounds for large component probabilities in multiple critical random graph models.

Findings

Polynomial upper bounds for large component probabilities

Applicable to intersection, percolation, and inhomogeneous graphs

Simplifies analysis of critical random graph behavior

Abstract

In this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models of random graphs when considered at criticality. Specifically, we apply our method to a model of random intersection graph, a random graph obtained through $p$-bond percolation on a general $d$-regular graph, and a model of inhomogeneous random graph.